Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19084 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,9,13)(5,14,17,18)(6,16,10,15)(7,12,8,11)(19,20,25,22)(21,29,32,34)(23,28,24,27)(26,33,30,31);(3,5)(4,10)(6,13)(7,15)(8,16)(9,17)(11,18)(12,14)(19,21,25,32)(20,26,22,30)(23,31,24,33)(27,29,28,34);(3,6,9,10)(4,5,13,17)(7,18,8,14)(11,16,12,15)(19,22,25,20)(21,30,32,26)(23,27,24,28)(29,33,34,31);(3,7,9,8)(4,11,13,12)(5,15,17,16)(6,14,10,18)(19,23,25,24)(20,27,22,28)(21,31,32,33)(26,29,30,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258694);